Random data Cauchy problem for the nonlinear Schr\"{o}dinger equation with derivative nonlinearity

TL;DR

This paper establishes almost sure local and global well-posedness, along with scattering results, for a derivative nonlinear Schrödinger equation with random initial data in certain Sobolev spaces, extending understanding of such equations below critical regularity.

Contribution

It proves well-posedness and scattering for the derivative nonlinear Schrödinger equation with random data below the scaling critical regularity, a novel result in this context.

Findings

Almost sure local well-posedness in specified Sobolev spaces.

Global well-posedness and scattering for small data.

Results hold for dimensions and nonlinearities satisfying certain conditions.

Abstract

We consider the Cauchy problem for the nonlinear Schr\"{o}dinger equation with derivative nonlinearity $(i\partial _t + \Delta ) u= \pm \partial (\overline{u}^m)$ on $\R ^d$, $d \ge 1$, with random initial data, where $\partial$ is a first order derivative with respect to the spatial variable, for example a linear combination of $\frac{\partial}{\partial x_1} , \, \dots , \, \frac{\partial}{\partial x_d}$ or $|\nabla |= \mathcal{F}^{-1}[|\xi | \mathcal{F}]$. We prove that almost sure local in time well-posedness, small data global in time well-posedness and scattering hold in $H^s(\R ^d)$ with $s> \max \left( \frac{d-1}{d} s_c , \frac{s_c}{2}, s_c - \frac{d}{2(d+1)} \right)$ for $d+m \ge 5$, where $s$ is below the scaling critical regularity $s_c := \frac{d}{2}-\frac{1}{m-1}$.